When you say she's got a red rear end, do you mean the cloaca, or her rump? If the cloaca's red I would assume it's from blood pressure in the area, which could signal an oncoming prolapse. I've never dealt with a prolapse, I think if you're covering the rather wide range of nutritional bases you won't have that, I've heard it's related to or caused by a lack of healthy oils. A lot of pelletized feeds don't have any natural oils in unadulterated form. Sudden onset of laying large eggs would obviously threaten any hen, probably. Did your younger girls start laying with wind-eggs? I think that's a natural mechanism for working them up to laying full size eggs.
They're forced into laying by a variety of means including increasing/lessening hours of light/dark, the feed types, and of course they're bred to lay early and lay even if they're not actually up to it, so they exhaust themselves and don't live as long and comfortably as most other chooks. They tend to be feed-inefficient, production layers and broilers do, for example one production bird can need three chicken's feeds per day, while not actually producing that quantity in egg or meat. The breed's inefficient digestive system accounts for a lot of wastage. To counter that they bred their bodies extra small and the egg size extra big so the feed is more likely to go to the eggs rather than the bird's flesh. I think such heavy production birds will be phased out in future, they're a false economy, like cows whose udders drag on the floor.
When the second of the three chickens died the remaining one seemed to go into mourning and stopped laying and I thought that it missed the company. Does that make sense?
Yes, they do stop laying sometimes if something distressing occurs. Chickens aren't meant to be solo.